Maths (university) · Part 09
Determinants, inverses and vector spaces
Inverting matrices and understanding the structure behind linear algebra.
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- DeterminantA single number computed from a square matrix. For [[a,b],[c,d]] it is ad - bc. It tells you how much the matrix scales areas.
- Cofactor expansionCompute a larger determinant by going along a row, multiplying each entry by the determinant of the smaller matrix left after crossing out its row and column. Signs alternate + - +.
- Inverse matrixThe inverse of A, written A^(-1), satisfies A times A^(-1) = I. For [[a,b],[c,d]] it is (1/(ad - bc)) times [[d,-b],[-c,a]].
- Solving with the inverseIf A x = b and A is invertible, then x = A^(-1) b. Multiply both sides on the left by the inverse.
- Vector spaceA set of vectors you can add together and scale by numbers, while always staying inside the set. A subspace is a vector space sitting inside a bigger one.
- Linear independenceVectors are independent if none of them can be built from the others by scaling and adding. Otherwise they are dependent.
- Span and basisThe span of vectors is everything you can build from them by scaling and adding. A basis is an independent set that spans the whole space.
- Rank and dimensionThe rank of a matrix is the number of independent columns. The dimension of a space is the number of vectors in a basis. Rank plus nullity equals the number of columns.
- Linear transformationA function on vectors that keeps addition and scaling intact. Every such function in coordinates is multiplication by a matrix.